Low Dimensional Quantum Magnetism in the Copper Oxides

Abstract

The magnetism of lamellar copper oxides, which are the parent materials of high temperature superconductors, is dominated by the spin 1/2 Cu+2 ions on the CuO2 planes. The magnetic behavior of these planes at high temperature is described well by the planar quantum Heisenberg antiferromagnetic (AFM) model, which has long range order only at T = 0. In fact they have three dimensional AFM order due to weak spin anisotropics and interplane couplings. Starting from a Hubbard model with spin orbit and Coulomb exchange couplings, we first derive an effective single-bond magnetic Hamiltonian which contains these anisotropics and couplings. Using a simple Coulomb interaction, as used by Moriya, reveals an interesting hidden symmetry which implies a rotationally symmetric interaction. For orthorhombic LCO, this symmetry is removed only when one adds the different bonds on the lattice, yielding a net Dzyaloshinskii-Moriya antisymmetric anisotropy. In tetragonal symmetry, the zero point quantum spin wave energy (QZPE) generates additional four-fold symmetry terms and delicate higher order interplane interactions, which help select a ground state among states which would otherwise be degenerate due to frustration. Adding several more interplanar interactions yields the full effective magnetic Hamiltonian, which is then used to identify the magnetic structures and competitions among them, leading to phase diagrams in parameter space. These are also used to discuss the critical phenomena which occur near various possible transitions. Specific attention will be devoted to the structures of tetragonal Sr2CuO2Cl2, Nd2CuO4 and Pr2CuO4 In the former, frustration among layers is lifted by pseudodipolar interactions and by QZPE. In the latter two, the rare earth also participates in the magnetism. Finally, we give full analysis of Sr2Cu3O4Cl2, which contains Cu3O4 planes with an extra Cu ion in the center of every second Cu plaquette. Each of the two types of Cu ions reaches AFM order separately, and the system also exhibits an interesting ferromagnetic moment. A theory in which the intersublattice coupling is described by pseudodipolar interactions allows us to deduce various coupling constants. Similarities in the relative Cu-O-Cu geometries enable us to relate these coupling constants to the nearest and next nearest neighbor interactions in many chain, ladder and lamellar cuprates. These have consequences concerning the magnetic behavior of the latter systems: competing nearest and next nearest neighbor couplings may explain the spin gap observed in the chains, and the new pseudodipolar interactions remove frustration in the interladder coupling.